Ocean weather forecasting

The field of physical oceanography has matured to a point where it is now conceivable to combine numerical models and observations via data assimilation in order to provide ocean predict

Ocean Weather Forecasting

An Integrated View of Oceanography

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ISBN-13 978-1-4020-3981-2 (HB)

ISBN-13 978-1-4020-4028-3 (e-book)

© 2006 Springer

eminent scientist in the domains of ocean physics, tides, satellite altimetry, and ocean modeling He was also a pioneer in the development of operational oceanography

Part I: Introduction

Chapter 1: N Smith, Perspectives from the Global Ocean Data

Part II: Modeling

Chapter 2: S Griffies, Some ocean models fundamentals 19

Chapter 3: A.M Tréguier, Models of ocean: Which ocean? 75 Chapter 4: R Bleck, On the use of hybrid vertical coordinates in

Chapter 5: E Blayo and L Debreu, Nesting ocean models 127

Part III: Oceanographic observations and atmospheric forcing

Chapter 6: I Robinson, Satellite measurements for operational

Chapter 7: U Send, In-situ observations: Platforms and techniques 191 Chapter 8: S Pouliquen, In-situ observations: Operational systems

Chapter 9: W Large, Surface fluxes for practitioners of global

Part IV: Data assimilation

Chapter 10: P Brasseur, Ocean data assimilation using sequential

Chapter 11: I Fukumori, What is data assimilation really solving,

and how is the calculation actually done? 317

Chapter 12: F Rabier, Importance of data: A meteorological

Chapter 13: D Anderson, M Balmaseda, and A Vidard, The

ECMWF perspective 361

Part V: Systems

Chapter 14: P Bahurel, MERCATOR OCEAN global to regional

ocean monitoring and forecasting 381

Chapter 15: M Bell, R Barciela, A Hines, M Martin, A Sellar,

and D Storkey, The Forecasting Ocean Assimilation Model

(FOAM) system 397

Chapter 16: E Chassignet, H Hurlburt, O.M Smedstad,

G Halliwell, P Hogan, A Wallcraft, and R Bleck, Ocean

prediction with the HYbrid Coordinate Ocean Model (HYCOM) 413

Chapter 17: A Schiller and N Smith, BLUElink: Large-to-coastal

scale operational oceanography in the Southern Hemisphere 427

Chapter 18: J.F Minster, Operational oceanography: A European

perspective 441

Chapter 19: Y Desaubies, MERSEA: Development of a European

ocean monitoring and forecasting system 449

Chapter 20: L Crosnier and C Le Provost, Internal metrics

definition for operational forecast systems inter-comparison:

Example in the North Atlantic and Mediterranean Sea 455

Chapter 21: J Harding and J Rigney, Operational oceanography

Chapter 22: M Altalo, Applications of ocean forecast information

for economic advancement in developed and developing societies 483

methods based on the Kalman filter 271

507

Chapter 24: A Oschlies, On the use of data assimilation in

Chapter 25: J Wilkin and L Lanerolle, Ocean forecast and

analysis models for coastal observatories 549

Index 575

Ø

the drift of objects and substances in the ocean

Chapter 23: B Hackett, Breivik and C Wettre, Forecasting

Progress in a wide range of ocean research and applications depends upon the prompt and dependable availability of ocean information products The field of physical oceanography has matured to a point where it is now conceivable to combine numerical models and observations via data assimilation in order to provide ocean prediction products on various spatial and time scales As a result, many nations have begun large-scale efforts to provide routine products to the oceanographic community The Global Ocean Data Assimilation Experiment (GODAE) provides a framework for these efforts, i.e., a global system of observations, communications, modeling, and assimilation that will deliver regular, comprehensive information on the state of the oceans, in a way that will promote and engender wide utility and availability of this resource for maximum benefit

to the community The societal benefit will be an increased knowledge of the marine environment and ocean climate, predictive skills for societal, industrial, and commercial benefit and tactical and strategic advantage, as well as the provision of a comprehensive and integrated approach to the oceans

We therefore considered it timely, given the international context, to bring together leading scientists to summarize our present knowledge in ocean modeling, ocean observing systems, and data assimilation to present

an integrated view of oceanography and to introduce young scientists to the current state of the field and to a wide range of applications This book is the end result of an international summer school held in 2004 that aimed, among other things, at forming and motivating the young scientists and professionals that will be the principal movers and users of operational oceanographic outputs in the next 10 years The chapters collected in this volume cover a wide range of topics and are authored not only by scientists, but also by system developers and application providers

ix

We would like to thank all the speakers for providing a stimulating series of lectures at this GODAE Summer School We also express our appreciation to the members of the scientific committee and to the GODAE IGST who contributed in numerous ways to the success of the school We thank all the attendees (see list in Appendix) for participating actively in the lecture review process and for creating a most cordial atmosphere We thank Jean-Michel Brankart, Laurence Crosnier, Nicolas Ferry, and David Rozier for preparing and putting together a superb set of student exercises Finally, our thanks go to Yves Ménard, Joëlle Guinle, Véronique Huix, Nicole Bellefond, and Josiane Brasseur who spent a considerable time with the logistics of the school before and after A special thank goes to Josiane Brasseur for her help in formatting the manuscripts

Primary support for this GODAE summer school was provided by the Centre National d’Etudes Spatiales (CNES), the MERSEA EU project, and GODAE Additional funding was provided by the National Science Foundation (NSF) and by the National Aeronautics and Space Administration (NASA) This support is gratefully acknowledged

PERSPECTIVES FROM THE GLOBAL OCEAN DATA ASSIMILATION EXPERIMENT

Neville Smith

Bureau of Meteorology Research Centre, Melbourne, Victoria, Australia

Abstract : The Global Ocean Data Assimilation Experiment (GODAE) is

introduced, including a discussion of the historical basis and conceptual framework GODAE aims to develop a global system of observations, communications, modeling and assimilation that will deliver regular, comprehensive information on the state of the oceans in a way that will promote and engender wide utility and availability of this resource for maximum benefit to society The overall strategy and guiding principles are introduced and the core components discussed The data and modeling and assimilation systems are intended to provide infrastructure serving a broad range of users and applications The targeted applications include open- ocean forecasts, coastal and regional prediction, climate assessments and prediction, and reanalyzes for scientific and other purposes Both internal and external metrics have been developed to assure the quality and reliability of the product streams The focus at present is on developing an understanding and more intimate relationship with the user community

Keywords : Ocean, data assimilation, observations, prediction

1 Introduction

The concept of a Global Ocean Data Assimilation Experiment (GODAE) emerged from the Ocean Observation Panel for Climate (OOPC) in 1997 and derived from concern that attracting the investment for an adequate long-term global ocean observing system depended upon a clear demonstration of the feasibility and value of such a system (Smith and Lefebvre, 1997) Using the First GARP (Global Atmospheric Research Program) Global Experiment (FGGE) as a model, OOPC proposed GODAE

as an experiment in which a comprehensive, integrated observing system would be established and held in place for several years and the data

1

E P Chassignet and J Verron (eds.), Ocean Weather Forecasting, 1-17

© 2006 Springer Printed in the Netherlands

assimilated into state-of-the art models of the global ocean circulation in near real-time to demonstrate utility

GODAE recognized the pioneering work in operational oceanography in the U.S (see Peloquin, 1992, and other papers within that special volume) and the fact that interest in building a broader global capability was emerging in several nations (for example, MERCATOR in France; Bahurel, this volume) This work, among others, guided the development of the concept and ultimately the strategy (International GODAE Steering Team (IGST), 2000) and implementation plan (http://www.bom.gov.au/GODAE/)

As with many international initiatives, GODAE by itself does not provide resources or develop capacity Rather it relies on the resources and capacity derived from national or regional initiatives and GODAE’s role is one of coordination and cooperation and, for example, introducing standards and references for the business of operational oceanography

This paper recounts the development of GODAE and some perspectives drawn from experience and from those who are thinking of the future of operational ocean analysis and prediction In order to provide a little context for GODAE in relation to the evolution of ocean science and the development of weather prediction, Section 2 discusses some historical aspects and section 3 some of the lessons learnt from numerical weather prediction Section 4 discusses the rationale and scope while section 5 introduces the core components Other chapters of this volume examine these components (e.g., observations, models, assimilation) in more detail Section 6 discusses applications and the utility of GODAE products and some of the issues surrounding the use of model products Again, there are several papers in this volume (e.g., Hackett et al.) that go into this area in more detail Section 7 discusses methods the GODAE community is using to test and validate their products and services Section 8 discusses the user community and implications for the systems and methods being developed within GODAE The final section provides some brief conclusions

2 A little history

Scientific observation of the oceans did not begin in earnest till about the nineteenth century; till this time, exploration and expanding ocean trade routes were the primary concern Advances in communication technology led to the idea of using under-sea cables to connect the American and European continents This required knowledge of the sea bed and thus led to exploration of the depth of the ocean; until this point, almost all knowledge

of the oceans was derived from surface observation Along with the improvements in knowledge of the depth of the sea, it was discovered that

life did exist at great depth Scientific cruises for systematic exploration

were born The British Challenger expedition from 1872 to 1876 and German exploration on the Gazelle from 1874 to 1876 were two of the early

successful deep sea expeditions, taking systematic measurements of ocean currents, temperature, chemistry and biology, as well as sampling bottom sediments

Valuable trading routes had been started on the open seas and travel time was a critical element of commercial success M.F Maury, superintendent of the Depot of Charts and Instruments at Washington, D.C., began to collect and collate information on surface currents and weather conditions leading

to the publication of The Physical Geography of the Sea (Maury 1859),

making it one of the first practical applications of ocean science and ocean observations If a point in time has to be chosen to mark the beginning of operational oceanography, this time is it Maury led the organization of an international system for regular observation; sailors on all vessels at sea would regularly record certain observations (e.g., sea state, sea surface temperature, weather, etc.) and, in exchange, they would be provided with charts of ocean currents and weather conditions in order to plan their voyage The legacy of these early efforts can still be appreciated in the GODAE systems of today

These scientific endeavors marked the start of what Neumann and Pierson (1966) termed the first era of oceanographic research The three-dimensional structure of the ocean was being observed for the first time The second era was born out of the realization that the ocean was not stationary and that its circulation could be partly explained by theoretical relationships (e.g., Ekman, 1905) Exploration of the oceans moved into the four-dimensional era; expeditions of the early twentieth century were making more accurate physical and chemical measurements and the station spacing was closer, driven in part by theoretical revelations While this era probably marked the first awareness of spatial and temporal sampling problems, it was to be many years later before the ramifications of aliasing and poor spatial resolution were to be fully appreciated

The third era was characterized by significant technological advances, such as the bathythermograph, and by highly organized, intensive oceanographic surveys which sought quasi-synoptic sampling of large regions This era also marked the introduction of non-ship instrumentation such as drifting and moored buoys One of the more imaginative innovations

of this period was the neutrally buoyant float (Swallow 1955), a technology that lies at the heart of the Argo campaign of today This period was also marked by significant advances in theory, not the least being the first theoretical explanations of the gyres and intense western boundary current depicted in Maury's chart (e.g., Stommel, 1948; Sverdrup 1947)

The modern era of oceanography has been shaped by at least three factors First, costs and logistical considerations have driven the development of mooring and autonomous underwater and surface technology These advances combined with real-time telemetry not only make synoptic observation of the ocean practical, but allow data to be delivered to models quickly

A second significant factor is satellites The vastness of the oceans has, and will forever, preclude near-simultaneous sampling of the oceans by

conventional, in situ instrumentation, even at the surface Remote sensing

offers the promise of ocean data over all regions of the globe, simultaneously, though restricted to a surface view

near-A third factor is related to both the previous factors - computing The growth in computational capacity over the last 50 years has been phenomenal For observationalists, it has revolutionized instrumentation, allowing more detailed and accurate recording and near-instantaneous processing, both on research ships and on moorings and autonomous devices, and in land-based laboratories Computing power was the key enabling technology for satellites Computers have revolutionized the capacity of ocean modelers to represent the circulation of the actual ocean It

is this capacity, as much as any other, which has underpinned the evolution

of modern oceanography to the point where routine, operational oceanography is feasible and the concept of GODAE, makes sense

The legacy from ocean research experiments such as the Tropical Ocean Global Atmosphere Experiment (TOGA; McPhaden et al.,1998) and the World Ocean Circulation Experiment (WOCE; e.g., Smith 2001) is also very important TOGA developed systematic observation and routine prediction

of seasonal-to-interannual climate variations (e.g., El Nino) with requirements closely related to those of GODAE and operational oceanography WOCE introduced many innovations in observation and developed the models and assimilation methods that are the basis for many GODAE systems

Perspective #1: Scientific and technical advances over the last century, including accrued knowledge of the dynamics and physics of the ocean, provide the basis for developing the systems of GODAE

3 Lessons from meteorology

At the First GODAE Symposium, Dr Tim Palmer delivered a lecture

“En Route to GODAE: A brief history of NWP” (see www.bom.gov.au/GODAE) and, within that lecture, he cited from Charney

et al (1969) concerning US participation in the then Global Atmospheric Research Program (GARP): “It is estimated that the data requirements of computer models are met for only 20 per cent of the earth’s surface Vast oceanic regions remain unobserved… the earth-orbiting satellite affords the opportunity of developing an economically feasible global observing capability.” Meteorologists were concerned with their ability to observe the relevant atmospheric variables, at all levels and globally, and to have that data available each day for models and forecasts Moreover, on the basis of progress made with atmospheric models, they wished to test the hypothesis that models and data assimilation could extend useful predictability and provide useful forecasts, at lead times several days ahead of what was possible at that time

The goals of GARP were effectively (a) deterministic weather forecasting and (b) understanding climate The First GARP Global Experiment (FGGE) was conceived to address the challenges above and set down several specific goals:

(i) Development of more realistic models for extended range

forecasting, general circulation studies, and climate

(ii) To assess the ultimate limit of predictability of weather systems

(iii) To develop more powerful methods for assimilation of

meteorological observations and, in particular, for using synchronous data

non-(iv) To design an optimum composite meteorological observing

system for routine numerical weather prediction

Bengtsson (1981) discusses the impact of FGGE on numerical weather prediction, the meteorological counterpart of the systems GODAE is developing It is clear that significant progress was made against each of the goals of FGGE and that that experiment was critical in the development of modern weather prediction systems Palmer also showed the evolution of forecast skill since FGGE, around 2 extra days in lead time in the Northern Hemisphere, and over 3 for the Southern Hemisphere This progress has been made possible by better observations (particularly remote sensing), better models, faster computers, and most importantly, a vastly improved knowledge of the dynamics and physics of the atmosphere The improved skill however only tells part of the story The information content of a modern numerical weather prediction system bears little resemblance to its predecessors during FGGE Regional models are often operating at scales of 5-10 km or better, and these broad measures of skill do not capture the immense value added through finer resolution (indeed, in some cases, the systems are penalized!) Many forecasts systems are also producing more than one forecast (ensembles) so that the users can now apply forecasts with knowledge of the probability of an event occurring Assimilation systems are

also being extended, for example to consider ozone, air quality and carbon dioxide Finally, these same systems are being used to produce consistent (re-)analyses of the atmospheric state

While there are significant differences between the goals of numerical ocean prediction (the GODAE focus) and numerical weather prediction, it is also clear that our community can benefit from the experiences of that community, including their failures We will discuss objectives and products that closely parallel those discussed here It is also likely GODAE systems will utilize and/or share a great deal of the infrastructure developed for weather prediction, including observational networks, data and product communication networks, computers and organizational infrastructure One difference that is worth considering is that at this time the numerical ocean prediction community does not have the benefit of a dedicated ocean research program GARP has morphed into the World Climate Research Program, which does consider climate aspects, but its Programmes do not provide the focus that we need now and in the future

Perspective #2: We have a good model to follow in

the development of numerical weather prediction and we

can deliver efficiency and effectiveness by partnering and

sharing with this community

4 The concept of and rationale for GODAE

The vision of GODAE is (IGST, 2000):

A global system of observations, communications, modeling and assimilation, that will deliver regular, comprehensive information on the state of the oceans in a way that will promote and engender wide utility and availability of this resource for maximum benefit to society

“

”

Regular depictions of the ocean state will be obtained through synthesis

of observations with ocean model estimates The models will allow us to assimilate and integrate complex information in a way that is consistent with our knowledge of ocean dynamics and physics

Scientifically, in the totality of its complexity, the problem is enormous Yet, it is evident that most aspects are now tractable The benefits of assimilation of ocean observations into ocean and climate models has been demonstrated (e.g., Ji et al., 1998; Giraud et al., 1997; Burnett et al., 2002, and papers within that Volume; Wunsch, 2001) A system of ocean data collection and modeling of the ocean that will allow us to follow the state of the ocean routinely seems in the realm of feasibility (see also Smith and Lefebvre, 1997)

4.2 The rationale

GODAE is inspired by both opportunity and need There is a genuine user demand for ocean products, for a range of time and space scales (e.g., Flemming 2001, Altalo, this Volume) There is also a concern for future ocean research A capability for providing regular ocean analyses is required

as a framework for scientific research and development In addition, if we are to build a future with a robust, routine, permanent and well-supported network of ocean observations, then a clear and convincing demonstration of the feasibility, practicality and value of maintaining such a network is required

The opportunities arise because of the development and maturity of remote and direct observing systems, making global real-time observation feasible; the steady advances in scientific knowledge and our ability to model the global ocean and assimilate data at fine space and time scales; the genuine enthusiasm of the ocean community to promote and implement integrated global observing systems; and the critical advances provided by research programs (see Section 2)

The underlying rationale for the organization of this activity as an international experiment is that achieving the GODAE vision will not happen serendipitously and that the needed capacity will not be realized without a concerted effort to ensure, first, proper integration of the components and, second, the commitment to proving value and viability

4.3 The approach

Smith (2000) and IGST (2000) introduce the objectives and scope of GODAE and the reader is referred to those publications and the GODAE web site for details

One premise is that GODAE is not just concerned with prediction in the traditional sense (looking forward in time), but prediction in its most general form, where information is extrapolated and interpolated in space and time, and between fields (Figure 1) The objectives intentionally imply a broad scope in the belief that wide utilization and exploitation of products are essential for cost-efficiency and relevance to society

Figure 1 Schematic of the processes used to exploit data In some cases we use linear,

perhaps empirical relationships to relate the current state to, say, a likely future state In other cases forecasts are produced based on current data (“today”), perhaps at a specific location (“here”), and perhaps for a subset of the total variable space (“ours”), in order to forecast the state in the future (“tomorrow”), at some remote location (“there”) or for some variables that are not part of the observables (“yours”) The process involves extrapolation (e.g., a forecast), interpolation (e.g., discrete points to a grid) and interpretation (e.g., inferring winds from sea surface topography)

The strategy for the development of these products is built on the concept

of a GODAE “Common” which is shared by all GODAE Partners responsible for realizing the goals and objectives of GODAE The GODAE Common concept is essential for GODAE, and must also be transported into the “operational” environment, for example through data policies and scientific cooperation

5 Building the systems

The essential building blocks of GODAE are observations, models and estimation tools (Figure 2) In the GODAE context, these elements are inextricably linked, with obvious two-way interdependencies The generation of globally consistent fields of ocean temperature, salinity and velocity components through the synthesis of multivariate satellite and in situ data streams into ocean models is a central theme of GODAE

Figure 2 Illustration of the process for taking in situ and remotely sensed data (left) through a

model-based assimilation system to produce a self-consistent analysis, which is then used to produce products such as a climate or regional/coastal forecast

The scope and international nature of GODAE requires distributed data assembly and serving, a multiplicity of assimilation products, distributed product serving and archiving, and a multiplicity of application centers (Figure 3)

5.1 GODAE observational and data needs

Data needed for GODAE model/assimilation systems can be separated into four main classes: atmospheric forcing (wind stress, wind speed, air temperature, specific humidity, precipitation) and sea-ice, data for assimilation (e.g., altimetry, Argo, SST), validation data (e.g., hydrography) and ancillary data (climatologies, bathymetry) Note, however, that the separation into data types is neither definitive nor unique (e.g., forcing data can be used as one of the controls on the assimilation process)

Koblinsky and Smith (2001) discusses the data system and other papers

of this Volume discuss details and issues that are of specific concern for GODAE Remote sensing data is naturally central to the success of GODAE and GODAE has placed particular emphasis on surface topography, surface wind and sea surface temperature data

GODAE itself has taken two specific initiatives to address specific gaps

In the early stages of GODAE it became clear that the in situ coverage was inadequate for both climate and ocean assimilation purposes The Argo Pilot Project (Argo Science Team, 1998) was established soon after GODAE was born, and has realized a near-revolution in our capability to observe the ocean in real-time (see papers by Send and by Pouliquen, this Volume) A

second Pilot Project arose in a somewhat unexpected area, sea surface temperature; a field that the community had believed was being estimated well The GODAE High-Resolution SST Pilot Project (see chapter by I Robinson in this volume) aims to deliver integrated, high-resolution products, derived from a range of different, but complementary observing systems, that properly respect our understanding of the near-surface temperature structure (e.g., the skin effect) and addresses issues such as the diurnal cycle

Various data servers will be responsible for maintaining and monitoring the data flow to assimilation groups and to those undertaking validation/evaluations The GODAE Monterey server and the CORIOLIS Centre (see chapter by S Pouliquen in this volume) are two examples of this important functionality One of the tasks is to link the server functions together so that the data users will have a consistent and transparent interface

to the variety of data that are available One of the challenges facing GODAE (and others) is the establishment of adequate metadata to facilitate data tracking, intercomparisons, and distribution of data which may undergo revision through various quality control procedures

Perspective #3: The real impact of GODAE will

come through its ability to bring its complex data and

information to applications and users

5.2 Models and data assimilation

Because of the irregular and incomplete nature of the datasets relative to the scales of interest, a considerable burden in ocean state estimation and forecasting is placed not only on the assimilation components but also on the model The model provides a capacity to extrapolate information, enabling past data to be used for present analyses, and present data to be used as a basis for predictions of the ocean state at future times (forecasts) Other papers in this Volume discuss approaches to modeling and data assimilation and some of the issues faced by the GODAE community

Most of the target applications require good representation of, at least, temperature and velocity components and sea level High resolution operational oceanography requires accurate depiction of mesoscale ocean features such as eddies and the meandering of currents and fronts and of upper ocean structure Coastal applications require accurate sea level and cross-shelf transport estimates Seasonal-to-interannual climate forecasts require a good representation of the upper ocean temperature and salinity fields Decadal climate monitoring and research requires attention to the thermohaline circulation, among other things Biogeochemical applications

require attention to the upper ocean physical parameterizations and the vertical transports (upwelling) All require considerable computational resources for global simulation and so rely on advanced software developments to take advantage of state of the art computer technology

Perspective #4: Global high-resolution ocean model assimilation systems are the main focus of GODAE Regional prototypes have proved critical for development and for regional applications Sector- specific systems (e.g., for global climate estimates) are also an important aspect Reanalyses are an important strategy

An outstanding issue for GODAE, with implications for assimilation and prediction, is the degree to which the key fields mentioned above are predictable and, secondly, the extent to which provided fields (boundary conditions, initial conditions, other inputs) in effect enhance predictability (skill) to the target systems The applied nature of GODAE only allows it to address these issues in passing, so again it is important that supporting research is fostered to test and understand all aspects of predictability Note that in the context of GODAE, such research applies not only to temporal predictions (forecasts) but to the more general context (see Fig 1)

The use of a variety of approaches to modeling and assimilation is regarded as a strength in the strategy of GODAE Within a framework of intercomparison and progressive evaluations, the diversity of approaches can

be used to quantify uncertainties and test reliability of ocean state estimates and initial conditions and forecasts

Perspective #5: The oceans are predictable … but when and where, and for how long? What are the dependencies and limitations? Observations? Representation of ocean dynamics and physics? Assimilation? Parameterizations? GODAE will provide only the first installment in our quest to address these issues

6 The utility of GODAE outputs

The key outcome will be significant improvement in the detail, quality, timeliness, availability and usefulness of ocean analysis and prediction products The reader is also referred to the GODAE Implementation Plan on http://www.bom.gov.au/GODAE/ for detail of activities by different groups and a more complete description of applications

Coherent, organized data sets: GODAE aims to develop more

coherent, better organized, more widely available and more useful data sets Such outputs will be realized through:

(a) More effective assembly and availability From the outset, the GODAE participants recognized that they must work to build coherent data streams that remove the mysteries associated with specific measurement techniques and the confounding problems associated with merging data of different types and formats

(b) Improving data utility GODAE places a high-premium on the wide use of data and products to ensure observing efforts realize their full potential in operational systems

(c) Improving data quality A sub-project has been launched to coordinate and standardize the GODAE approach (see www.bom.gov.au/GODAE/) As operational oceanography systems mature, they will provide routine, regular and immediate testing of data and thus add value to data sets

These outputs depend upon adequate devotion of effort to all stages of data handling Efficiency is realized through rationalization and streamlining

of the procedures

Reanalyses and synoptic ocean analyses: GODAE is most readily

associated with products of ocean model assimilation, usually in the form of space-time gridded fields GODAE includes the continual revision and improvement of analyses, either through re-analysis or through intercomparison activities The great worth of reanalyzes lies in the fact that they provide dynamically and physically consistent estimates over a period,

in a form that is readily used by research, but also by the broader marine community who have interests (dependencies) on knowledge of ocean variability and predictability

Short-range ocean forecasts: GODAE will have a leading role in

short-range ocean prediction and a supporting role in coupled air-sea prediction and surface wind waves via the provision of related ocean fields to application centers While we might argue that 4-dimensional assimilation is

at its roots simply a means for projecting and synthesizing data in space and time, the capacity to extend this projection (initial condition) forward in time

to produce forecasts gives the system special value

Climate applications: The most common application for the GODAE

ocean state estimate is as an initial condition for a coupled model forecast (e.g., Ji et al., 1998) One of the primary issues to be faced by this community is how best to use the state estimate; for example, the nature of

the problem might favor an ensemble of initial conditions rather than a single, high-fidelity product For decadal variability and longer-term change, GODAE focuses on the provision of consistent, high-quality analyses and re-analyses of the ocean

Coastal applications: Coastal applications will use GODAE ocean state

estimates as boundary conditions for coastal/ littoral zone hindcasts and forecasts and analyses (Figure 3)

Figure 3 A schematic of the hypothetical nesting of a coastal application near Sydney within

the BLUElink ocean forecasting system (indicated by fine dots; see Schiller and Smith, this volume) or a coarser seasonal prediction system “POAMA” (Wang et al., 2001).

It is not yet clear what the accuracy requirements are Development of GODAE products for these applications will represent a significant research effort within the community Issues of nesting of models of different resolution, the importance of regional wave and surge models, consistency

in bathymetry, forcing, boundary configurations, and input to ecosystem models are critical elements for collaboration The end users will include regional/local governments responsible for coastal management, as well as coastal industries such as fishing and recreation

The GODAE approach provides efficiency because the systems can provide information/boundary conditions to multiple users, in a variety of ways In some prototypes the regional/local modeling is in-built to the modeling system In other cases the coastal modeling is part of the same project so the interface issues are being solved as part of the project In yet other cases, boundary conditions are being provided to third parties who

e utility of GODAE pro

7 How to measure success?

The demonstration of the quality and value of GODAE products for res

ific rationale for, and a more detailed description of the GO

Perspective #6: Implementing a rigorous system of internal and

may have knowledge of the source model (and vice versa) but otherwise are running completely independent systems/applications

There are a large number of issues that impact th

ducts We are for the present slave to the errors of our atmospheric partners Accurate ocean surface current predictions and simulations may prove as elusive as atmospheric fluxes and winds, and we do not yet fully understand the extent to which subsurface currents can be predicted We do not yet know how well can we “predict” boundary conditions for coastal applications, and how much it matters when we get it “right”

earch and operational applications is the central objective of the experiment We need to set standards for data and products that are testable and defensible There are two levels of evaluation criteria Internal (technical) evaluation criteria should measure the performance of the components and functions, effectively within the GODAE Common External measures and feedback will come from GODAE users and applications

The scient

DAE metrics are given in Le Provost et al (2002) The internal metrics will include measures of consistency, quality and performance/efficiency The so- external evaluation criteria include (a) the impact of GODAE products for the different applications, (b) the utility of GODAE products for the research community, (c) the number of users and their level of satisfaction, (d) the extent of resultant innovation, (e) the utility for observing system evaluation and design, and (f ) the extent of uptake by value-adders and other specific users

external tests and in tercomparisons in order to evaluate systems and

to set standards is a key task We need to foster the development of international infrastructure, and national infrastructure, to support and monitor the performance and effectiveness of systems

8 Users and benefits

At least four types of relationships with end users have been identified

needs are satisfied by directly utilizing the products and services Direct to the Public This suits the ad hoc and occasional user whos

needed for the

DAE faces a challenge of determining the apabilities and product availability relevant to the areas and to build a con

ing the utility of products for different

emanating from GODAE Centers or application centers There is no intermediary or down-stream value-adding

Via middle-users/value-adders In this case specialists, varying from private ocean enterprises to sector-specific

perhaps after blending it with other information and/or rendering it in a form that is more useful and consumable , provide it to their clients The middle-users have expertise from both the provider and client sides and value-adding is through a partnership

Direct to specific users/sectors In some cases, specific users may be able to directly exploit GODAE product

commercial Value-adding is entirely on the user side

Capacity building and education Here the users do not have access to sophisticated systems or technology and support is

Perspective #7: Determin

users and sectors of the ocean community is the major challenge at this time

9 nclusions

system and in the global and regional operational oceanographic systems tha

gic goals The former rep

Co

t are being developed and tested now and that we envisage being maintained by several nations GODAE has achieved a level of investment that exceeded its expectations but such investments will only be sustained through proving the utility and use of GODAE deliverables and offset by tangible economic and social returns and outcomes

Like weather prediction, GODAE contains a balance between the practical and applied and the long-term strate

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SOME OCEAN MODEL FUNDAMENTALS

NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey, USA

Abstract The purpose of these lectures is to present elements of the equations and

algorithms used in numerical models of the large-scale ocean circulation Such models generally integrate the ocean's primitive equations, which are based on Newton's Laws applied to a continuum fluid under hy- drostatic balance in a spherical geometry, along with linear irreversible thermodynamics and subgrid scale (SGS) parameterizations During formulations of both the kinematics and dynamics, we highlight issues related to the use of a generalized vertical coordinate The vertical co- ordinate is arguably the most critical element determining how a model

is designed and applications to which a model is of use

Keywords: Ocean modelling, parameterization, vertical coordinate

Numerical ocean models are computational tools used to understand and predict aspects of the ocean They are a repository for our best ocean theories, and they provide an essential means to probe a mathe- matical representation of this very rich and complex geophysical system That is, models provide an experimental apparatus for the scientific rationalization of ocean phenomena Indeed, during the past decade,

oceanographers and climate scientists The reason for this state of affairs

is largely due to improved understanding of both the ocean and ocean models, as well as increased computer power allowing for increasingly realistic representations of ocean fluid dynamics Without computer models, our ability to develop a robust and testable intellectual basis for ocean and climate dynamics would be severely handicapped

19

E P Chassignet and J Verron (eds.), Ocean WeatherForecasting, 19-73

O 2006 Springer Printed in the Netherlands

The remainder of this section introduces some basic concepts, themes,and questions, some of which are revisited later in the lectures Wepresent some philosophical notions which motivate a focus on funda-mental concepts and notions when designing, constructing, and analyz-ing ocean models.

1.1 Model environments

The field of ocean model design is presently undergoing a rapid growthphase It is arguable that the field has reached adolescence, with furthermaturation likely taking another 10-20 years as we take the models to

a new level of integrity and innovation Many applications drive thisevolution, such as studies of climate change, operational oceanography,and ultra-refined resolution process studies

One goal of many developers is that the next decade of model tion will lead to a reduction in code distinctions which presently hinderthe ability of modelers to interchange algorithms, make it difficult todirectly compare and reproduce simulations using different codes, andincrease the burdens of model maintenance in a world of increasinglycomplex computational platforms and diverse applications Notably,the distinctions will not be removed by all modelers using a common al-gorithm Such is unreasonable and unwarranted since different scientificproblems call for different algorithmic tools Instead, distinctions may

evolu-be removed by the development of new codes with general algorithmicstructures flexible enough to encompass multiple vertical coordinates,different horizontal grids, various subgrid scale (SGS) parameterizations,and alternate numerical methods

The word environment has recently been proposed to describe thesehighly flexible and general codes As yet, no model environment exists

to satisfy the needs and desires of most modelers Yet some models aremoving in this direction by providing the ability to choose more thanone vertical coordinate This is a critical first step due to the centralimportance of vertical coordinates The present set of lectures formulatesthe fundamental equations using generalized vertical coordinates, andthese equations form the basis for generalized vertical coordinate oceanmodels Ideally, the advent of general model environments will allowscientists to use the same code, even though they may use differentvertical coordinates, horizontal grids, numerical methods, etc

Many of the ideas presented here are an outgrowth of research anddevelopment with the Modular Ocean Model of Griffies et al., 2004, aswell as the MITgcm (Marshall et al., 1997, Adcroft and Campin, 2004).The MITgcm provides for a number of depth-based and pressure-based

vertical coordinates Another approach, starting from an isopycnal ered model, has been taken by the Hybrid Coordinate Ocean Model(HYCOM) of Bleck, 2002 HYCOM is arguably the most mature of thegeneralized vertical coordinate models.

lay-From an abstract perspective, it is a minor point that different elers use the same code, since in principle all that matters should be thecontinuum equations which are discretized This perspective has, un-fortunately, not been realized in practice Differences in fundamentals

mod-of the formulation and/or numerical methods mod-often serve to make thesimulations quite distinct, even when in principle they should be nearlyidentical Details do matter, especially when considering long time scaleclimate studies where small differences have years to magnify

An argument against merging model development efforts is that there

is creative strength in diversity, and so there should remain many oceancodes A middle ground is argued here, whereby we maintain the frame-work for independent creative work and innovation, yet little effort iswasted developing redundant software and/or trying to compare differ-ent model outputs using disparate conventions To further emphasizethis point, we stress that the problems of ocean climate and operationaloceanography are vast and complex, thus requiring tremendous humanand computational resources This situation calls for merging certainefforts to optimize available resources Furthermore, linking modelerstogether to use a reduced set of code environments does not squelch cre-ativity nor does it lead to less diversity in algorithmic approaches In-stead, environments ideally can provide modelers with common startingpoints from which to investigate different methodologies, parameteriza-tions, and the like

The proposal for model environments is therefore analogous to use of afew spoken/written languages (e.g., english, french) to communicate andformulate arguments, or a few computer languages (e.g., Fortran, C++)

to translate numerical equations into computer code Focusing on a fewocean model environments, rather than many ocean models, can lead toenhanced collaboration by removing awkward and frustrating barriersthat exist between the presently wide suite of model codes Ultimately,such will (it is hoped!) lead to better and more reproducible simulations,thus facilitating the maturation of ocean modelling into a more robustand respectable scientific discipline

1.2 Some fundamental questions

It is possible to categorize nearly every question about ocean elling into three classes

mod-1 Questions of model fundamentals, such as questions raised in thissection.

2 Questions of boundary fluxes/forcing, from either the surface sea, river-sea, and ice-sea interactions, or forcing from the solidearth boundary The lectures in this volume from Bill Large touchupon many of the surface flux issues

air-3 Questions of analysis, such as how to rationalize the simulation toenhance ones ability to understand, communicate, and conceptu-alize

If we ask questions about physical, mathematical, or numerical aspects

of an ocean model, then we ask questions about ocean model tals The subject deals with elements of computational fluid mechanics,geophysical fluid mechanics, oceanography (descriptive and dynamic),and statistical physics Given the wide scope of the subject, even amonograph such as Griffies, 2004 can only provide partial coverage Weconsider even less in these lectures The hope is that the material willintroduce the reader to methods and ideas serving as a foundation forfurther study

fundamen-For the remainder of this section, we summarize a few of the manyfundamental questions that designers and users often ask about oceanmodels Some of the questions are briefly answered, yet some remainunaswered because they remain part of present day research It is no-table that model users, especially students learning how to use a model,often assume that someone else (e.g., their adviser, the author of a re-search article, or the author of a book) has devoted a nontrivial level ofthought to answering many of the following questions This is, unfor-tunately, often an incorrect assumption The field of ocean modelling

is not mature, and there are nearly as many outstanding questions asthere are model developers and users Such hopefully will provide mo-tivation to the student to learn some fundamentals in order to help thefield evolve

Perhaps the most basic question to ask about an ocean model concernsthe continuum equations that the model aims to discretize

Should the model be based on the non-hydrostatic equations, asrelevant for simulations at spatial scales less than 1km, or is the hy-drostatic approximation sufficient? Global climate models have allused the hydrostatic approximation, although the model of Mar-shall et al., 1997 provides an option for using either Perhaps in10-20 years, computational power will be sufficient to allow fullynon-hydrostatic global climate simulations Will the simulations

change drastically at scales larger than 1km, or do the hydrostaticmodels parameterize non-hydrostatic processes sufficiently well formost applications at these scales? Note that the accuracy of thehydrostatic approximation scales as the squared flow aspect ratio(ratio of vertical to horizontal length scales) Atmospheric mod-elers believe their simulations will be far more realistic with anexplicit representation of non-hydrostatic dynamics, such as con-vection and cloud boundary layer processes In contrast, it remainsunclear how necessary non-hydrostatic simulations are for globalocean climate Perhaps it will require plenty of experience run-ning non-hydrostatic global models before we have unambiguousanswers.

Should the kinematics be based on incompressible volume serving fluid parcels, as commonly assumed for ocean models us-ing the Boussinesq approximation, or should the more accuratemass conserving kinematics of the non-Boussinesq fluid be used, ascommonly assumed for the more compressible atmosphere Oceanmodel designers are moving away from the Boussinesq approxi-mation since only a mass conserving fluid can directly representsea level changes due to steric effects (see Section 3.4.3 of Griffies,2004), and because it is simple to use mass conserving kinematics

con-by exploiting the isomorphisms between depth and pressure cussed by DeSzoeke and Samelson, 2002, Marshall et al., 2003, andLosch et al., 2004

dis-Can the upper ocean surface be fixed in time with a rigid lid, asproposed decades ago by Bryan, 1969 and used for many years, orshould it be allowed to fluctuate with a more realistic free surface

so to provide a means to pass fresh water across the ocean surfaceand to represent tidal fluctuations? Most models today employ afree surface in order to remove the often unacceptable restrictions

of the rigid lid Additionally, many free surface methods removeelliptic problems from hydrostatic models The absence of ellipticproblems from the free surface models greatly enhances their com-putational efficiency on parallel computers (Griffies et al., 2001).Should tracers, such as salt, be passed across the ocean surface viavirtual tracer fluxes, as required for rigid lid models, or should themodel employ real water fluxes thus allowing for a natural dilutionand concentration of tracer upon precipitation and evaporation, re-spectively? As discussed more fully in Section 3.6, the advent offree surface methods allows for modelers to jettison the unphysical

virtual tracer methods of the rigid lid Nonetheless, virtual tracerfluxes remain one of the unnecessary legacy approximations plagu-ing some modern ocean models using free surface methods Thepotential problems with virtual tracer fluxes are enhanced as thetime scales of the integration go to the decade to century climatescale.

What is the desired manner to write the discrete momentum tion: advective, as commonly done in B-grid models, or vector in-variant, as commonly in C-grid models? The answer to this ques-tion may be based more on subjective notions of elegance thanclear numerical advantage

equa-How accurate should the thermodynamics be, such as the equation

of state and the model’s “heat” tracer? The work of McDougalland collaborators provides some guidance on these questions (Mc-Dougall, 2003, McDougall et al., 2003, Jackett et al., 2004) Howimportant is it to get these things accurate? The perspective takenhere is that it is useful to be more accurate and flexible with presentday ocean climate models, since the temperature and salinity rangeover which they are used is quite wide, thus making the older ap-proximations less valid Additionally, many of the more accurateapproaches have been refined to reduce their costs, thus makingtheir use nearly painless

After deciding on a set of model equations, further questions ariseconcerning how to cast the continuum partial differential equations onto

a finite grid First, we ask questions about the vertical coordinates.Which one to use?

Geopotential (z-coordinate): This coordinate is natural for nesq or volume conserving kinematics and is most commonly used

Boussi-in present-day global ocean climate models

Pressure: This coordinate is natural for non-Boussinesq or massconserving kinematics and is commonly used in atmospheric mod-els As mentioned earlier, the isomorphism between pressure anddepth allow for a straightforward transformation of depth coordi-nates to pressure coordinates, thus removing the Boussinesq ap-proximation from having any practical basis We return to thispoint in Section 6

Terrain following sigma coordinates: This coordinate is commonlyused for coastal and estuarine models, with some recent effortsaimed as using it for global modelling (Diansky et al., 2002)

Potential density or isopycnal coordinates: This coordinate is monly used for idealized adiabatic simulations, with increasing usefor operational and global climate simulations, especially whencombined with pressure coordinates for the upper ocean in a hybridcontext.

com-Generalized hybrid vertical coordinates: Models formulated forgeneral vertical coordinates allow for different vertical coordinatesdepending on the model application and fluid regime Models withthis facility provide an area of focus for the next generation ofocean models

What about the horizontal grid? Although horizontal grids do notgreatly determine the manner that many physical processes are repre-sented or parameterized, they greatly influence on the representation ofthe solid-earth boundary, and affect details of how numerical schemesare implemented

Should we cast the model variables on one of the traditional Athrough E grids of Arakawa and Lamb, 1977? Which one? The

B and C grids are the most common in ocean and atmosphericmodelling Why? Section 3.2 of Griffies et al., 2000a providessome discussion of this question along with references

What about spectral methods commonly used in atmospheric els? Can they be used accurately and effectively within the com-plex geometry of an ocean basin? Haidvogel and Beckmann, 1999present a summary of these methods with application to the ocean.Typically, spectral methods have not been useful in the horizontalwith realistically complex land-sea boundaries, nor in the verticalwith realistically sharp pycnoclines The reason is that a spectralrepresentation of such strong gradients in the ocean can lead to un-acceptable Gibbs ripples and unphysically large levels of spuriousconvective mixing

mod-Should the horizontal grid cells be arranged according to sphericalcoordinates, even when doing so introduces a pesky coordinatesingularity at the North Pole? What about generalized orthogonalcoordinates such as a bipolar Arctic coupled to a spherical regionsouth of the Arctic (Figure 1)? Such grids are very common today

in global modelling, and their use is straightforward in practicesince they retain the regular rectangular logic assumed by sphericalcoordinate models Or what about strongly curved grid lines thatcontour the coast, yet remain locally orthogonal? Haidvogel and

Beckmann, 1999 provide some discussion of these grids and their uses

What about nested regions of refined resolution where it is critical

to explicitly resolve certain flow and/or boundary features? Blayo

at this school (see also Blayo and Debreu, 1999) illustrates the potentials for this approach Can it be successfully employed for long term global climate simulations? What about coastal impacts

of climate change? These are important questions at the forefront

of ocean climate and regional modelling

Can a non-rectangular mesh, such as a cubed sphere, be success- fully used to replace all coordinate singularities with milder sin- gularities that allow for both atmosphere and ocean models to

a compelling case for this approach, whereby both the ocean and

provides a schematic of a cubed-sphere tiling of the sphere What about icosahedrons, or spherical geodesics as invented by Buckminster filler? These grids tile the sphere in a nearly isotropic manner Work at Colorado State University by David Randall and collaborators has shown some promise for this approach in the at- mosphere and ocean

What about finite element or triangular meshes popular in engi- neering, tidal, and coastal applications? These meshes more ac- curately represent the solid earth boundary Or what about time dependent adaptive approaches, whereby the grid is refined ac- cording to the time dependent flow regimes? Both methods have traditionally failed to perform well for realistic ocean climate simu- lations due to problems representing stratified and rotating fluids However, as reported in this volume by Jens Schroter, some im- portant and promising advances have been made by researchers

at the University of Reading and Imperial College, both in Eng- land, as well as the Alfred-Wegener Institute in Germany Their efforts have taken strides in overcoming some of the fundamental problems If this area of research and development is given time

to come to fruition, then perhaps in 10 years we will see finite ele-

lPolar filtering is a method to reduce the spatial scales of the simulation as one approaches the coordinate singularity a t the North Pole Many computational and numerical problems have been encountered with this approach

ments commonly used for regional and global models Such couldrepresent a major advance in ocean modelling.

Figure 1 Illustration of the bipolar Arctic as prescribed by Murray, 1996 (see his Figure 7) and realized in the global model discussed in Griffies et al., 2005 A similar grid has also been proposed by Madec and Imbard, 1996 Shown here are grid lines which are labeled with the integers for the grid points The grid has 360 points in the generalized longitude direction, and 200 points in the generalized latitude direction This, or similar, bipolar Arctic grids are commonly used in global ocean modelling to overcome problems with the spherical coordinate singularity at the North Pole Note that the cut across the Arctic is a limitation of the graphics, and does not represent

a land-sea boundary in the model domain.

Figure 2 Cubed sphere tiling of the sphere Note the singularities at the cube corners are much milder than a spherical coordinate singularity found with spherical grids at the poles The cubed sphere tiling has been implemented in the MITgcm for both the atmosphere and ocean model components This figure was kindly provided

by Alistair Adcroft, a developer of the MITgcm.

What processes are represented explicitly, and what are the impor- tant ones to parameterize? This is one of the most critical and difficult questions of ocean model design and use The lectures by Anne Marie Treguier from this school summarizes many of the issues She notes that the choice of model resolution and parameterization prejudices the simulation so much so that they effectively determine the "ocean" to

be simulated Discussions in Chassignet and Verron, 1998 thoroughly survey various aspects of the parameterization problem This book is from a 1998 school on ocean modelling and parameterization Many of the issues raised there are still unresolved today Finally, Griffies, 2004 has much to say about some of the common parameterizations used in ocean climate models

Numerical methods are necessary to transform the continuum equa- tions into accurate and efficient discrete equations for stepping the ocean forward in time There are many methods of use for doing this task Should they be based on finite volume methods? Such methods are becoming more common in ocean modelling They provide the numericist with a useful means to take the continuum equations and cast them onto a finite grid

What sorts of time stepping schemes are appropriate, and what properties are essential to maintain? Will the ubiquitous leap-frog methods2 be supplanted by methods that avoid the problematic time splitting mode? Chapter 12 of Griffies, 2004 provides a dis- cussion of these points, and argues for the use of a time staggered method, similar to that discussed by Adcroft and Campin, 2004 and used in the Hallberg Isopycnal Model (Hallberg, 1997) and Modular Ocean Model version 4 (Griffies et al., 2004)

Should the numerical equations maintain a discrete analog to con- servation of energy, tracer, potential vorticity, and potential en- strophy satisfied by the ideal continuum equations? For long term climate simulations, tracer conservation is critical What about the other conserved quantities?

What are the essential features needed for the numerical tracer advection operator? Should it maintain positivity of the tracer

ily realized in their adjoint form as required for 4D variational

2As noted in Griffies et al., 2000a, the majority of ocean models supported for large-scale oceanography continue to use the leapfrog discretization of the time tendency

assimilation (see the lectures at this school from Jens Schr¨oter aswell as Thuburn and Haine, 2001).

How should the model treat the Coriolis force? On the B-grid, it

is common to do so implicitly or semi-implicitly in time, but thismethod is not available on the C-grid since the velocity componentsare not coincident in space Also, the C-grid spatial averaging ofthe Coriolis force can lead to problematical null modes (Adcroft

et al., 1999)

What about the pressure gradient calculation? We return to thisquestion in Section 5, where comments are made regarding thedifficulties of computing the pressure gradient

1.3 Two themes

There are two themes emphasized in these lectures

How the vertical coordinate is treated is the most fundamentalelement of an ocean model design

The development of ocean model algorithms should be based onrational formulations starting from fundamental principles.The first theme concerns the central importance of vertical coordinates

in ocean model design Their importance stems from the large tions at present between algorithms in models with differing verticalcoordinates Further differences arise in analysis techniques These fun-damental and pervasive distinctions have led to disparate research anddevelopment communities oriented around models of a particular class ofvertical coordinate One purpose of these lectures is to describe methodswhereby these distinctions at the formulation stage are minimized, thus

distinc-in prdistinc-inciple facilitatdistinc-ing the design of a sdistinc-ingle code capable of employdistinc-ingmany vertical coordinates

The second theme is a “motherhood” statement What scientist orengineer would disagree? Nonetheless, it remains nontrivial to satisfy forthree reasons First, there are many important elements of the oceanthat we do not understand This ignorance hinders our ability to pre-scribe rational forms for the very important SGS operators Second,some approximations (e.g., Boussinesq approximation, rigid lid approx-imation, virtual tracer fluxes), made years ago for good reasons then,often remain in use today yet need not be made with our present-daymodelling capabilities and requirements These legacy approximationsoften compromise a model’s ability to realistically simulate certain as-pects of the ocean and/or its interactions with other components of the

climate system Third, developers are commonly under intense time

hoe measures which, unfortunately, tend to stay around far longer than

originally intended

In our presentation of ocean model fundamentals, we find it useful to start with a discussion of fluid kinematics Kinematics is that area of me- chanics concerned with the intrinsic properties of motion, independent

of the dynamical laws governing the motion In particular, we establish expressions for the transport of fluid through a specified surface The specification of such transport arises in many areas of oceanography and ocean model design

There are three surfaces of special interest in this section

The lower ocean surface which occurs at the time independent solid earth boundary This surface is commonly assumed to be impenetrable to fluid.3 The expression for fluid transport at the lower surface leads to the solid earth lcinematzc boundary condition

To formulate budgets for mass, tracer, and momentum in the ocean, we consider the upper ocean surface to be a time dependent permeable membrane through which precipitation, evaporation, ice melt, and river runoff pass The expression for fluid transport

at the upper surface leads to the upper ocean lcinematzc boundary

condition

portance when establishing the balances of mass, tracer, and mo- mentum within a layer of fluid whose upper and lower bounds are

Mass conservation for an infinitesimal parcel of fluid means that as it moves through the fluid, its mass is constant in time

3 ~ h i s assumption may be broken in some cases For example, when the lower boundary is

a moving sedimentary layer in a coastal estuary, or when there is seeping ground water We

do not consider such cases here

In this equation, A 4 = pdV is the parcel's mass, p is its in situ den- sity, and dV is its infinitesimal volume The time derivative is taken following the parcel, and is known as a material or Lagmngian time

appropriate for the sphere, such as those illustrated in Figures 1 and 2 (see chapters 20 and 21 of Griffies, 2004 for a presentation of generalized horizontal coordinates)

For many purposes in fluid mechanics as well as ocean model design,

it is useful to transform the frame of reference from the moving parcel to

a fixed point in space This transformation takes us from the material

or Lagrangian frame to the Eulerian frame It engenders a difference in how observers measure time changes in a fluid parcel's properties In particular, the material time derivative picks up a tmnsport or advective term associated with motion of the parcel

This relation allows us to write the Lagrangian

conservation in an Eulerian conservation form4

P,t + V ( p v ) = 0

(3) expression (2) for mass

(4)

Fluids that conserve mass are said to be compressible since the vol- ume of a mass conserving fluid parcel can expand or contract based on pressure forces acting on the parcel, or properties such as temperature and salinity However, in many circumstances, it is useful to consider the kinematics of a parcel that conserves its volume, in which case

parcel's velocity that must be satisfied at each point of the fluid Fluid

4Throughout these lectures, a comma is used as a shorthand for partial derivative Hence,

physics It is a useful means t o distinguish a derivative from some of the many other uses of subscripts, such as a tensor component or as part of the name of a variable such as the fresh water flux q, introduced in equation (27)